Finding Cliques of a Graph using Prime Numbers

We present a prototype of a software tool for exploration of multiple combinatorial optimisation problems in large real-world and synthetic complex networks. Our tool, called GraphCombEx (an acronym of Graph Combinatorial Explorer), provides a unified framework for scalable computation and presentation of high-quality suboptimal solutions and bounds for a number of widely studied combinatorial optimisation problems. Efficient representation and applicability to large-scale gr...

It is shown how to construct a clique graph in which properties of cliques of a fixed order in a given graph are represented by vertices in a weighted graph. Various definitions and motivations for these weights are given. The detection of communities or clusters is used to illustrate how a clique graph may be exploited. In particular a benchmark network is shown where clique graphs find the overlapping communities accurately while vertex partition methods fail.

Finding a Maximum Clique is a classic property test from graph theory; find any one of the largest complete subgraphs in an Erd\"os-R\'enyi G(N, p) random graph. We use Maximum Clique to explore the structure of the problem as a function of N, the graph size, and K, the clique size sought. It displays a complex phase boundary, a staircase of steps at each of which 2log2 N and Kmax, the maximum size of a clique that can be found, increases by 1. Each of its boundaries has a fi...

In this paper, we study the maximum clique problem on hyperbolic random graphs. A hyperbolic random graph is a mathematical model for analyzing scale-free networks since it effectively explains the power-law degree distribution of scale-free networks. We propose a simple algorithm for finding a maximum clique in hyperbolic random graph. We first analyze the running time of our algorithm theoretically. We can compute a maximum clique on a hyperbolic random graph $G$ in $O(m + ...

In this short paper, following the most recent advances in complex network theory, a new approach to number theory with potential applications to other fields is proposed. The model by Garcia-Perez, Serrano and Boguna, introduces an algorithm which allows to create a bipartite graph of integers (with primes and composites) statistically very close to the real one. Since the algorithm is defined a priori, we can have a description of the simulated prime number distribution in ...

Maximum Flow Network Interdiction Problem (MFNIP) is known to be strongly NP-hard problem. We solve a simple form of MFNIP in polynomial time. We review the reduction of MFNIP from the clique problem. We propose a polynomial time solution to the Clique Problem.

We consider the maintenance of the set of all maximal cliques in a dynamic graph that is changing through the addition or deletion of edges. We present nearly tight bounds on the magnitude of change in the set of maximal cliques, as well as the first change-sensitive algorithms for clique maintenance, whose runtime is proportional to the magnitude of the change in the set of maximal cliques. We present experimental results showing these algorithms are efficient in practice an...

Identifying cliques in dense networks remains a formidable challenge, even with significant advances in computational power and methodologies. To tackle this, numerous algorithms have been developed to optimize time and memory usage, implemented across diverse programming languages. Yet, the inherent NP-completeness of the problem continues to hinder performance on large-scale networks, often resulting in memory leaks and slow computations. In the present study, we critically...

Finding the clique of maximum cardinality in an arbitrary graph is an NP-Hard problem that has many applications, which has motivated studies to solve it exactly despite its difficulty. The great majority of algorithms proposed in the literature are based on the Branch and Bound method. In this paper, we propose an exact algorithm for the maximum clique problem based on the Russian Dolls Search method. When compared to Branch and Bound, the main difference of the Russian Doll...

Finding maximum cliques in large networks is a challenging combinatorial problem with many real-world applications. We present a fast algorithm to achieve the exact solution for the maximum clique problem in large sparse networks based on efficient graph decomposition. A bunch of effective techniques is being used to greatly prune the graph and a novel concept called Complete-Upper-Bound-Induced Subgraph (CUBIS) is proposed to ensure that the structures with the potential to ...